The following two diagrams are commutative :
T o r A ( P , M ) → ψ ( S , R ) H ( S ⊗ A R ) T o r A ( g , f ) ↓ ↓ H ( g ~ ⊗ f ~ ) T o r A ( P ′ , M ′ ) → ψ ( S ′ , R ′ ) H ( S ′ ⊗ A R ′ ) ,
\begin{array}{ccc}
\mathrm{Tor}^A(P, M) & \xrightarrow{\psi(S, R)} & \mathrm{H}(S \otimes_A R) \\
\mathrm{Tor}^A(g, f) \downarrow & & \downarrow \mathrm{H}(\tilde{g} \otimes \tilde{f}) \\
\mathrm{Tor}^A(P', M') & \xrightarrow{\psi(S', R')} & \mathrm{H}(S' \otimes_A R') ,
\end{array}
Tor A ( P , M ) Tor A ( g , f ) ↓ Tor A ( P ′ , M ′ ) ψ ( S , R ) ψ ( S ′ , R ′ ) H ( S ⊗ A R ) ↓ H ( g ~ ⊗ f ~ ) H ( S ′ ⊗ A R ′ ) , H ( H o m g r A ( R ′ , E ′ ) ) → φ ( R ′ , E ′ ) E x t A ( M ′ , N ′ ) H ( H o m g r A ( f ~ , h ~ ) ) ↓ ↓ E x t A ( f , h ) H ( H o m g r A ( R , E ) ) → φ ( R , E ) E x t A ( M , N ) .
\begin{array}{ccc}
\mathrm{H}(\mathrm{Homgr}_A(R', E')) & \xrightarrow{\varphi(R', E')} & \mathrm{Ext}_A(M', N') \\
\mathrm{H}(\mathrm{Homgr}_A(\tilde{f}, \tilde{h})) \downarrow & & \downarrow \mathrm{Ext}_A(f, h) \\
\mathrm{H}(\mathrm{Homgr}_A(R, E)) & \xrightarrow{\varphi(R, E)} & \mathrm{Ext}_A(M, N) .
\end{array}
H ( Homgr A ( R ′ , E ′ )) H ( Homgr A ( f ~ , h ~ )) ↓ H ( Homgr A ( R , E )) φ ( R ′ , E ′ ) φ ( R , E ) Ext A ( M ′ , N ′ ) ↓ Ext A ( f , h ) Ext A ( M , N ) . Let α : L ( M ) → R , α ′ : L ( M ′ ) → R ′ , γ : L ( P ) → S , γ ′ : L ( P ′ ) → S ′ \alpha : L(M) \to R, \alpha' : L(M') \to R', \gamma : L(P) \to S, \gamma' : L(P') \to S' α : L ( M ) → R , α ′ : L ( M ′ ) → R ′ , γ : L ( P ) → S , γ ′ : L ( P ′ ) → S ′
a ∘ α = p M , a ′ ∘ α ′ = p M ′ , b ∘ γ = p P , b ′ ∘ γ ′ = p P ′ .
a \circ \alpha = p_M, \quad a' \circ \alpha' = p_{M'}, \quad b \circ \gamma = p_P, \quad b' \circ \gamma' = p_{P'} .
a ∘ α = p M , a ′ ∘ α ′ = p M ′ , b ∘ γ = p P , b ′ ∘ γ ′ = p P ′ . By definition, H ( g ~ ⊗ f ~ ) ∘ ψ ( S , R ) \mathrm{H}(\tilde{g} \otimes \tilde{f}) \circ \psi(S, R) H ( g ~ ⊗ f ~ ) ∘ ψ ( S , R )
H ( g ~ ⊗ f ~ ) ∘ H ( γ ⊗ α ) = H ( ( g ~ ∘ γ ) ⊗ ( f ~ ∘ α ) ) ,
\mathrm{H}(\tilde{g} \otimes \tilde{f}) \circ \mathrm{H}(\gamma \otimes \alpha) = \mathrm{H}((\tilde{g} \circ \gamma) \otimes (\tilde{f} \circ \alpha)) ,
H ( g ~ ⊗ f ~ ) ∘ H ( γ ⊗ α ) = H (( g ~ ∘ γ ) ⊗ ( f ~ ∘ α )) , whereas ψ ( S ′ , R ′ ) ∘ T o r A ( g , f ) \psi(S', R') \circ \mathrm{Tor}^A(g, f) ψ ( S ′ , R ′ ) ∘ Tor A ( g , f )
H ( γ ′ ⊗ α ′ ) ∘ H ( L ( g ) ⊗ L ( f ) ) = H ( ( γ ′ ∘ L ( g ) ) ⊗ ( α ′ ∘ L ( f ) ) ) .
\mathrm{H}(\gamma' \otimes \alpha') \circ \mathrm{H}(L(g) \otimes L(f)) = \mathrm{H}((\gamma' \circ L(g)) \otimes (\alpha' \circ L(f))) .
H ( γ ′ ⊗ α ′ ) ∘ H ( L ( g ) ⊗ L ( f )) = H (( γ ′ ∘ L ( g )) ⊗ ( α ′ ∘ L ( f ))) . On the other hand, α ′ ∘ L ( f ) \alpha' \circ L(f) α ′ ∘ L ( f ) f ~ ∘ α \tilde{f} \circ \alpha f ~ ∘ α L ( M ) L(M) L ( M ) R ′ R' R ′ a ′ ∘ ( α ′ ∘ L ( f ) ) = p M ′ ∘ L ( f ) = f ∘ p M = f ∘ a ∘ α = a ′ ∘ ( f ~ ∘ α ) a' \circ (\alpha' \circ L(f)) = p_{M'} \circ L(f) = f \circ p_M = f \circ a \circ \alpha = a' \circ (\tilde{f} \circ \alpha) a ′ ∘ ( α ′ ∘ L ( f )) = p M ′ ∘ L ( f ) = f ∘ p M = f ∘ a ∘ α = a ′ ∘ ( f ~ ∘ α ) α ′ ∘ L ( f ) \alpha' \circ L(f) α ′ ∘ L ( f ) f ~ ∘ α \tilde{f} \circ \alpha f ~ ∘ α γ ′ ∘ L ( g ) \gamma' \circ L(g) γ ′ ∘ L ( g ) g ~ ∘ γ \tilde{g} \circ \gamma g ~ ∘ γ ( γ ′ ∘ L ( g ) ) ⊗ ( α ′ ∘ L ( f ) ) (\gamma' \circ L(g)) \otimes (\alpha' \circ L(f)) ( γ ′ ∘ L ( g )) ⊗ ( α ′ ∘ L ( f )) ( g ~ ∘ γ ) ⊗ ( f ~ ∘ α ) (\tilde{g} \circ \gamma) \otimes (\tilde{f} \circ \alpha) ( g ~ ∘ γ ) ⊗ ( f ~ ∘ α )
H ( g ~ ⊗ f ~ ) ∘ ψ ( S , R ) = H ( ( g ~ ∘ γ ) ⊗ ( f ~ ∘ α ) ) = H ( ( γ ′ ∘ L ( g ) ) ⊗ ( α ′ ∘ L ( f ) ) )
H(\tilde{g} \otimes \tilde{f}) \circ \psi(S, R) = H((\tilde{g} \circ \gamma) \otimes (\tilde{f} \circ \alpha)) = H((\gamma' \circ L(g)) \otimes (\alpha' \circ L(f)))
H ( g ~ ⊗ f ~ ) ∘ ψ ( S , R ) = H (( g ~ ∘ γ ) ⊗ ( f ~ ∘ α )) = H (( γ ′ ∘ L ( g )) ⊗ ( α ′ ∘ L ( f ))) = ψ ( S ′ , R ′ ) ∘ T o r A ( g , f ) .
= \psi(S', R') \circ \mathrm{Tor}^A(g, f) .
= ψ ( S ′ , R ′ ) ∘ Tor A ( g , f ) . One reasons analogously for the second diagram.
Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.
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